Lemma 2.6 ([1]}, Lemma 2.5 in [2]}) 

Let \(\theta ^i_{jk}(u)\)  be the inner angle at \(i\)  in triangle \(\triangle ijk\)  in \((T,u*l)_E\) , and \(K(u)\in \mathbb {R}^V\)  is the discrete curvature. Then
\(&\frac{\partial \theta _{jk}^i}{\partial u_j} = \frac{\partial \theta _{ik}^j}{\partial u_i}\\=&\frac{r_i^2r_j^2\sin ^2 \Theta _{ij} + r_i^2r_jr_k(\cos \Theta _{jk}+ \cos \Theta _{ij}\cos \Theta _{ik}) + r_ir_j^2r_k(\cos \Theta _{ik} + \cos \Theta _{ij}\cos \Theta _{jk})}{l_{ik}l_{ij}^3\sin \theta _{jk}^i}.\) 
